Short-depth syndrome extraction circuits for calderbank shor steane (css) stabilizer codes

ABSTRACT

A disclosed methodology for syndrome extraction in a quantum measurement circuit includes generating a graph representing a code implemented by the quantum measurement circuit. The graph includes bit nodes corresponding to data qubits in the quantum measurement circuit, check nodes corresponding to syndrome qubits in the quantum measurement circuit, and edges between the bit nodes and check nodes that are each associated with a stabilizer measurement provided by the code. The methodology provides for assigning each of the different edges in the graph to a select one of “G” number of different edge types and performing at least G-number of temporally-separated rounds of qubit operations that each enact concurrent multi-qubit operations on endpoints of a subset of the edges assigned to a same one of the G different edge types.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. provisional applicationNo. 63/130,115, entitled “Short-Depth Syndrome Extraction Circuits forQuantum Codes,” and filed on Dec. 23, 2020, which is hereby incorporatedby reference for all that it discloses or teaches.

BACKGROUND

The scalability of decoders for quantum error correction is an ongoingchallenge in generating practical quantum computing devices. Hundreds orthousands of high-quality qubits with a very low error rate (e.g., 10⁻¹⁰or lower) may be needed to implement quantum algorithms with industrialapplications. Using current quantum technologies, these specificationscannot be met without using thousands of high-quality qubits that areeach individually encoded in thousands of physical qubits such thatthere may exist millions of qubits running each computation of thequantum computer. Obtaining error rates currently required by industrialapplications requires correcting, at regular intervals, errors thataccumulate over these millions of qubits. Detecting and correcting theseerrors entails processing a massive amount of data, leading tosignificant challenges in bandwidth and hardware resource allocation.

Calderbank Shor Steane (CSS) stabilizer codes are a special type of codeconstructed from classical codes with some special properties.Low-Density Parity-Check (LDPC) codes area one class of CSS codes thatshow promising results and could significantly reduce the overheadrequired for fault-tolerant quantum computation. However, as the errorcorrection code capability increases along with code depth, so tootypically does the processing time for measuring error syndromes. Moreefficient mechanisms for quantum LDPC code implementation are sought.

SUMMARY

According to one implementation, a method for extracting a syndrome froma quantum measurement circuit includes generating a graph representing aCSS code implemented by the quantum measurement circuit, where the graphincludes bit nodes corresponding to data qubits in the quantummeasurement circuit, check nodes corresponding to syndrome qubits in thequantum measurement circuit, and edges between the bit nodes and checknodes that are each associated with a stabilizer measurement provided bythe code. The method further provides for assigning each of thedifferent edges in the graph to a select one of “G” number of differentedge types and performing at least G-number of temporally-separatedrounds of qubit operations that each enact concurrent multi-qubitoperations on endpoints of a subset of the edges assigned to a same oneof the G different edge types. This methodology allows the syndrome tobe extracted with a constant depth circuit regardless of the particularCSS code employed.

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter.

Other implementations are also described and recited herein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example quantum computing system that implements ashort-depth syndrome extraction circuit that is of a constant depth forany CSS stabilizer code.

FIG. 2 illustrates an Tanner graph useful in illustrating error decodingprinciples for a quantum circuit implementing a CSS stabilizer code.

FIG. 3 illustrates an example traditional methodology for measuring anindividual stabilizer in a quantum circuit implementing a CSS stabilizercode.

FIG. 4A illustrates an exemplary Tanner graph useful for illustratingprinciples of syndrome extraction.

FIG. 4B illustrates exemplary aspects of a circuit compressionmethodology that can be applied to the Tanner graph of FIG. 4A to yielda short-depth syndrome extraction circuit.

FIG. 5 introduces another exemplary notation for a Tanner graph usefulin illustrating a methodology for extracting a syndrome from afully-connected quantum circuit implementing a hypergraph (HPG) productcode.

FIG. 6 illustrates an exemplary construction of a product graphrepresenting an HPG code formed by multiplying together two linear CSScodes.

FIG. 7A illustrates an example partial construction of another HPG codeproduct graph that illustrates a method for simultaneously measuringx-stabilizers and z-stabilizers.

FIG. 7B illustrates subgraph structure of the example HPG code productgraph shown in FIG. 7A.

FIG. 8 illustrates an example methodology for re-writing a CSS linearcode according to a balanced cardinal ordering scheme that permits afurther reduction in depth of an associated syndrome extraction circuit.

FIG. 9 illustrates example operations for measuring a syndrome in aquantum circuit with fully connected qubits implementing a CSSstabilizer code.

FIG. 10 illustrates example operations for measuring a syndrome in aquantum circuit with fully connected qubits implementing an HPG code.

FIG. 11 illustrates an exemplary computing environment suitable forimplementing aspects of the disclosed technology.

DETAILED DESCRIPTION

The herein disclosed technology proposes a short-depth syndromeextraction circuit for measuring error affecting quantum measurements incircuits that implement CSS codes. Measuring error in quantum circuitsmay entail what is referred to as “syndrome extraction” or themeasurement of an array of qubits that provide information about thelocation of faults (qubit errors) that have occurred during quantumcomputations. Syndrome extraction is performed via a mechanism known asthe “syndrome extraction circuit,” which refers to a sequence of quantumoperations (e.g., gates or joint parity measurements) that arecollectively effective to extract the syndrome. Using currentlyavailable syndrome extraction circuits, the “depth” of the syndromeextraction circuit—e.g., the number of time-separated quantum operationsor groups of such operations requisite to extract a syndrome—typicallygrows in proportion to the correction power of the particular errorcorrection code employed. There herein disclosed technology provides anovel syndrome extraction methodology that works for any CSS code andthat uses a smaller number of rounds of qubit operations than otherexisting approaches.

FIG. 1 illustrates an example quantum computing system 100 thatimplements a short-depth syndrome extraction circuit that is of constantdepth for any CSS stabilizer code. The quantum computing system 100includes a controller 102 that performs calculations by manipulatingqubits within a qubit register 108. The controller 102 includes logicfor executing quantum algorithms via such manipulations and forcontrolling a readout device 112 to extract information about thelocations of faults (errors) affecting qubit states during quantummeasurements. Specifically, the controller 102 includes short-depthsyndrome extraction logic 120 for generating control signals to commandthe readout device 112 to repeatedly extract an error syndrome (array ofbits and zeros) by implementing a syndrome measurement circuit 114within the readout device 112.

The syndrome measurement circuit 114 enables fault-tolerant quantumcomputing by applying a stabilizer code to the qubits in the qubitregister 108. Since measurement is known to destroy the delicate statesof a qubit needed for computation, the syndrome measurement circuit 114uses redundant qubits—known as “ancilla data bits” to performcomputations. During quantum processing, entropy from the data qubitsthat encode the protected data is transferred to the ancilla qubits thatcan be discarded. The ancilla qubits are positioned to interact withdata qubits such that it is possible to detect errors by measuring theancilla qubits and to correct such errors using a decoding unit 116 thatincludes one or more decoders. In some implementations, the decodingunit 116 includes logic executed by one or more classical computingsystems.

The syndrome measurement circuit 114 performs measurements of theancilla bits in the quantum computer to extract syndromes providinginformation measured with respect to errors (faults). In order to avoidaccumulation of errors during the quantum computation, the syndrome datais constantly measured, producing r syndrome bits for each syndromemeasurement round. In one implementation, the syndrome data is measuredwith a frequency of every 1 μs. Other implementations may measure thesyndrome more frequently, such as every 1 ns.

The repeatedly-measured syndrome data is used to identify and tracklocations of faults that occur throughout a quantum operation that spansseveral individual qubit manipulations. At the termination of thequantum operation, the measurement circuit performs a final measurementthat disentangles one qubit from the others in the syndrome measurementcircuit 114, and this qubit is read out as a final solution (a 1 or 0value). By using the syndrome data to track faults in each round of thequantum operation, a classical bit correction can be performed tocorrect the final solution.

The syndrome measurement circuit 114 extracts each round of the syndromedata by performing sequences of operations known as “stabilizermeasurements.” In the case of CSS codes, stabilizer measurements aretypically implemented with a particular defined sequence of operations.For example, a CSS code stabilizer may be measured by entangling anancilla qubit with a group of data qubits and by subsequently measuringthe ancilla qubit to observe the resulting state. This entanglementforces the group of neighbor data qubits into an eigenstate of astabilizer operator (e.g., the X-stabilizer or the Z-stabilizer),allowing one to measure the stabilizers without perturbing the system.When the stabilizer measurement outcomes change, this corresponds to oneor more qubit errors in the quantum state that are projected by themeasurement. The outcome is either 0 (trivial) or 1. If the outcome of ameasurement is 1, this indicates the presence of an error on the dataqubits measured. An ancilla qubit that supports a stabilizer measurementin this way is also referred to herein as a “check qubit.”

CSS code stabilizer measurements typically entail several time-separatedmeasurement steps, such as those exemplary steps disclosed herein withrespect to FIG. 3 below. Traditional solutions typically provide forperforming a different, time-separated stabilizer measurement withrespect to each check bit within each stabilizer measurement round. Fora robust CSS code employing a large number of stabilizers, this resultsin a syndrome measurement circuit of considerable depth (long temporallength to implement).

In contrast to these traditional larger-depth stabilizer circuits, thesyndrome measurements circuit 114 implements what is referred to hereinas a “short-depth” syndrome extraction circuit, meaning that thesyndrome measurement circuit 114 can extract each round of syndrome datafrom the CSS code using a fewer number of time-separated measurementsteps that traditional solutions. According to one implementation, thesyndrome measurement circuit 114 implements a stabilizer measurementthat is mathematically proven to be of minimal depth and also to workfor any CSS code.

At each round of syndrome measurement, the syndrome data is sent to thedecoding unit 116, which implements decoding algorithms to analyze thesyndrome data and to detect the location of each error and to correcteach error on the data qubits.

In decoding applications, a “connectivity graph” G=(Q, E) may be used tospecify the hardware layout without the qubit register 108, where eachvertex q ∈ Q corresponds to a qubit. In the presently-disclosedapplications, it is assumed that the selected quantum architectureallows for implementation of any Clifford gate, Pauli state preparation,and Pauli measurement on each individual qubit q ∈ Q. It is furtherassumed that some entangling operation, such as a CNOT gate or a jointmeasurement, is possible between a pair of qubits q and q′ if and onlyif they are connected by an edge in the graph, i.e., {q, q′} ∈ E. Inthis work, the case considered is one where there exist quantumarchitectures and corresponding graphs with highly connected qubits.“Highly connected” implies that each individual qubit is directlyconnected to a large number, such as a majority, of qubits in a qubitgrid. In contrast, local connectivity restricts multi-qubit operationsto pairs of nearest neighbor qubits.

This means that a quantum operation can be performed on any set ofqubits in the qubit register 108 without performing interim entanglementoperations to establish/entanglement between lines of qubits.

To implement a stabilizer code of block length N in the proposedhardware, more than just N physical qubits are employed. Rather, thecode state is stored in a subset of N of the qubits in the connectivitygraph, which are referred to herein as data qubits. In contrast to dataqubits, the above-mentioned “ancilla qubits” make up a second set ofphysical qubits that are used to measure the value of the stabilizers.Ancillas used to measure X-type errors are referred to herein as “X-typecheck qubits” while ancillas used to measure Z-type errors are referredto herein as “Z-type check qubits.”

To implement logic for measuring stabilizers, it is often useful toconsider a specific mapping of a type of graph known as the Tanner graphof a code to physical qubits in the connectivity graph usingcoordinates. An exemplary Tanner is graph is discussed below withrespect to FIG. 2.

FIG. 2 illustrates an Tanner graph 200 useful in illustrating errordecoding principles for a quantum circuit implementing a CSS stabilizercode. The quantum circuit is said to have a connectivity graph(representing qubit connectivity) with coordinates that correspond tothe Tanner graph 200 such that each vertex in the Tanner graph 200 maybe understood as corresponding to a qubit at a particular coordinate inthe connectivity graph. This assignment of coordinates to the verticesof the Tanner graph 200 is sometimes referred to as “standardcoordinates.”

The Tanner graph 200 is exemplary of one of many different types of CSSstabilizer codes suitable for implementing the herein disclosedshort-depth syndrome extraction methodology. In this example, theparticular code illustrated by the Tanner graph 200 is a Steane codethat includes check qubits represented by squares, data qubitsrepresented by circles, and connections therebetween represented byedges. Since an actual implementation of the syndrome measurementcircuit may include more than 15,000 qubits, the Tanner graph of FIG. 2is representative of a small portion of the circuit that includes sevendata qubits, three X-basis check qubits (cx1, cx2, cx3) and threeZ-basis check qubits (cz1, cz2, cz3). The herein disclosed methodologiesfor measurement of an X-basis syndrome are the same or similar to thedisclosed methodologies for measuring the Z-basis syndrome. For thisreason, the following example describes a stabilizer measurement withrespect to only an X-basis check qubit.

Measurement of a stabilizer is performed by executing a syndromeextraction circuit, which may be understood as decomposed into asequence of steps S₀,S₁, . . . S_(τ−1). A step consists of a set ofoperations allowed by the connectivity graph where each operation in theset has disjoint support and therefore can be implementedsimultaneously. In general, two or more qubit operations may beperformed on a single step (simultaneously) provided the two operationsdo not occur on a same edge of the associated connectivity graph (orTanner graph, which has edges mirroring those of the circuit'sconnectivity graph). The size of a syndrome extraction circuit is thetotal number of operations (including idle operations within thecircuit), while the depth of syndrome extraction circuit referred to thenumber of time steps T required to implement the circuit.

Measuring an X-basis syndrome of the portion of the circuit representedby the Tanner graph 200 entails (1) preparation of the X-basis checkqubits in a known state; (2) operations to entangle each one of theX-basis check qubits with the neighboring data qubits to which theX-basis check qubit is connected by edges in the Tanner graph 200; and(3) measuring each of the X-basis qubits following these entanglementoperations. Performing the above-enumerated operations with respect tothe three X-basis check qubits (cx1, cx2, and cx3) is equivalent tomeasuring three stabilizers (executing three stabilizer circuits) toextract a 3-bit syndrome, where a 0 bit indicates an absence of fault oran even-number of faults on the neighboring data qubits and a 1 bitindicates an odd number of faults on the neighboring data qubits.Complete syndrome extraction typically entails repeating theabove-described operations with respect to the Z-basis check qubits.

FIG. 3 illustrates an example traditional methodology 300 for measuringan individual stabilizer in a quantum circuit implementing a CSSstabilizer code. As noted above, extraction of a round of syndrome dataentails performing a different stabilizer measurement with respect toeach different X-basis and Z-basis check qubit in the measurementcircuit.

In some implementations, a stabilizer supported on a set of qubits canbe measured by using a single measurement ancilla qubit. FIG. 3illustrates a specific example of this, whereby a CSS code stabilizermeasurement is performed by entangling a check qubit with its fournearest-neighbor qubits and by the observing the resulting state of thecheck qubit. Traditionally, this stabilizer measurement is performed byimplementing a sequence of operations in a very particular order. Thissequence of operations includes preparing the ancilla (the check qubit)by initializing this qubit ground state; performing four sequential(time-separated) CNOT operations, where the CNOTs target the fournearest-neighbor data qubits with the check qubit acting as the control;and finally, performing a projective measurement of the resultingeigenstate ({circumflex over (X)}_(a), {circumflex over (X)}_(b),{circumflex over (X)}_(c), {circumflex over (X)}_(d)) by measuring thecheck qubit. This single individual stabilizer measurement consists of 6sequential measurements. Each stabilizer within the CSS code is measuredindependently.

To measure the syndrome of the three X-basis check qubits shown in FIG.2 (above), this process entails 18 total time-separated measurementsteps (e.g., 6 steps for each one of the three stabilizer measurements).Notably, this methodology provides for a number of measurement stepsthat is proportional to the number of check qubits in a given circuit.

Notably, the CNOT-based stabilizer readout strategy requires the abilityto perform a CNOT between the measurement ancilla qubit and each dataqubit in the support of the operator. However, this may not be possibledepending on the connectivity graph corresponding to the particularquantum hardware employed. Another strategy for implementing astabilizer measurement is therefore to use extra ancillas in addition tothe readout ancilla qubit to help with the measurement using a cat statecircuit. Since the herein-contemplated solutions consider quantumarchitectures with full qubit connectivity, these architectures relyingon the cat state circuit for stabilizer measurement are consideredexternal to the scope of this disclosure.

The methodology discussed below with respect to the following figuresconsiders syndrome extraction circuits for CSS codes in the setting offully connected qubits using CNOT-based stabilizer readout.

FIG. 4A illustrates an exemplary Tanner graph 400 for implementing ashort-depth syndrome extraction circuit that improves upon themethodology discussed above with respect to FIG. 3. Operations discussedbelow with respect to both of FIG. 4A and 4B provide syndrome extractionfor any CSS code according to a mathematically-guaranteed, shortestpossible total number of measurement operations (e.g., the syndromemeasurement circuit is of optimal depth).

FIG. 4A illustrates a Tanner graph 400 implementing an example CSS code,known as the Steane code, where the data bits and ancilla bits of thesyndrome measurement circuit are represented by corresponding nodes inthe Tanner graphs 502, 504. Specifically, the Tanner graphs 502, 504include data bit nodes (q1-q7) representing the data qubits, check nodes(cx1, cx2, cx3) representing ancilla qubits used to implement theX-basis stabilizer measurement (also referred to herein as X-basis checkqubits), and edges between representing qubit connectivity supported bythe quantum hardware that is utilized in the execution of a syndromeextraction circuit.

To specify the operations of the syndrome extraction circuit, eachoperation can be labeled by an integer corresponding to a time step atwhich it is applied. According to one implementation, the hereindisclosed short-depth syndrome extraction circuit for CSS codes is aCNOT circuit composed of state preparations, single-qubit measurements,and CNOT gates.

In particular, the syndrome circuit acts on data qubits and ancilla(readout) qubits which respectively correspond to the data bit nodes(q1-q7) representing the data qubits, check nodes (cx1, cx2, and cx3) inthe Tanner graph 400. For each check node (cx1, cx2, cx3) there exists acorresponding preparation and a measurement operation that occurs in theshort-depth syndrome measurement circuit. For each edge (e.g., an edge402) in the Tanner graph 400, there is a CNOT in the short-depthsyndrome measurement circuit between the qubits corresponding to thenodes connected by the edge (e.g., the data qubit corresponding to q1and the check qubit correspond to cx1). According to one implementation,a syndrome extraction circuit can be implemented described below (1-3),which is the same or similar to methods discussed above with respect toFIG. 2-3. Following this is a discussion of a transformation thateffectively shortens the depth of the syndrome extraction circuit.

A sequential syndrome extraction circuit corresponding to the layout ofthe Tanner graph 400 can be implemented in the following steps:

1. Prepare check qubits (corresponding to cx1, cx2, cx3) in theappropriate state during the 0 time step;

2. Taking each of the check qubits one at a time, apply its associatedCNOT gates sequentially leading to each CNOT being applied during adifferent time step in {1, 2, . . . τ−1} (e.g., for check qubit cx1,CNOT gates are applied on q1, q3, q5, q6 during 4 different sequentialtimesteps); and

3. Measure all check qubits in the appropriate basis during the finaltimestep {τ}. Notably, this methodology provides for sequentialapplication of the CNOT gates on each of the check qubits. For example,step (2) above would provide for preparation of the cx1 CNOTs, then forpreparation of the cx2 CNOTs, then for preparation of the cx3 CNOTs(note: this could be achieved in any order for cx1, cx2, cx3).

Below, discussed with respect to FIG. 4B, is a first circuittransformation that effectively compresses the preparations andmeasurements of the check qubits in every syndrome measurement applied,for any CSS code implemented by a quantum architecture. For example,step 2 in the above could apply the sequence of four CNOTs on cx1, cx2,and cx3, simultaneously rather than one at a time (resulting in fourtime-steps instead of twelve).

FIG. 4B illustrates aspects of a circuit compression methodology thatcan be applied to the Tanner graph 400 of FIG. 4A to yield a short-depthsyndrome extraction circuit. Although the disclosed methodology can beapplied in any fully-connected qubit architecture implementing a CSScode, FIG. 4B illustrates the same Tanner graph 400 that is describedabove with respect to FIG. 4A.

Given a syndrome extraction circuit such as that described above withrespect to FIG. 4A, an equivalent circuit is obtained by performing amodification that temporally compresses the syndrome extraction processdescribed above with respect to FIG. 4. Specifically, this methodologyentails a relabeling of the commuting CNOTs. The only logicalrestriction on such relabeling is that two CNOTs with overlappingsupport are not to be applied during the same time step. Stateddifferently, it is not permissible to simultaneously perform multipleCNOT operations that act on a same qubit. For example, it is notpermissible to simultaneously implement a CNOT on {q1, cx1} and {q3,cx1} because both affect the state of cx1.

To obtain a minimum-depth circuit satisfying the above constraint, asubgraph structure is considered—referred to herein as an adjacencygraph G(C) 408. Each qubit in the adjacency graph 408 corresponds to avertex in G(C) and each CNOT in this subcircuit corresponds to an edgein G(C) between the pair of vertices associated with the qubits that itis applied on. The ordering of the CNOTs performed with respect to anindividual check qubit (e.g., cx1) is referred to herein as an “edgecoloring scheme.” The terms “edge coloring” and “edge type” are usedinterchangeably herein to refer to an assignment that is given to eachedge that dictates the time step in which the associated edge CNOT is tobe applied.

In a given edge coloring scheme, a “color” is assigned to each edge suchthat no two edges connected to a same check node (vertices cx1, cx2 orcx3) have the same color. Since the figures are shown in black andwhite, these color assignments are also referred to herein as “edgetypes” and are depicted by lines of different styles rather than ofdifferent color, as shown in key 410. Thus, the terms “edge type” and“edge color” are intended to mean the same thing and are usedinterchangeably throughout.

The chromatic number of a graph is the minimum number of colors requiredto obtain an edge coloring. For bipartite graphs, it is well known thatthe chromatic number is equal to the maximum degree of the graph andthere exist efficient algorithms to compute edge colorings.

Using a coloring scheme illustrated in FIG. 4B, the syndrome extractioncircuit for the Tanner graph 400 can be compressed (time-wise) to allowfor some simultaneous CNOT operations on different check bits withoutaltering the circuit outcome. Specifically, the syndrome extractioncircuit described with respect to FIG. 4A (e.g., measuring eachstabilizer, one at a time) is equivalent to one in which CNOTs of thedifferent stabilizer circuits for the different check qubits aresimultaneously applied on edges assigned to the same color (edge type)during the same time step. This general concept is formally stated belowas Circuit Transformation 1 and is referred to periodically throughoutthe remainder of this disclosure:

-   -   Circuit Transformation 1: Given a subcircuit C consisting only        of pairwise commuting CNOTs and a coloring of the adjacency        graph G(C), an equivalent circuit is obtained by applying all        those CNOTs which correspond to edges with the same color during        the same time step.

To implement an improved short-depth syndrome extraction circuitutilizing Circuit Transformation 1 (above), a classical controllerinitially determines the number of “G” different edge types to assign tothe various edges within the quantum circuit, where “G” represents amaximum number of edges abutting any individual node of the Tanner graph400 implementing the CSS code on which the syndrome extraction circuitis to be implemented. In the Tanner graph 400 of FIG. 4B, each checknode has four edges, so there are four different edge types (Type A, B,C, and D) in the selected coloring scheme. Each edge is assigned to anedge type according to one simple rule: no two edges of a same edge typemay connect to the same check qubit.

After the paths are assigned, the x-syndrome of the data qubits can becomputed per the following operations. First, ancilla qubits (checkbits) are prepared in the |+

state. This preparation of all ancillas is performed in a same time step(e.g., time step 0).

Then, for each of the next G different time steps, a set of operationsis performed at each time on a different one of the edge types. In theexample of FIG. 4A, the forgoing is implemented by performing a firstset of CNOT operations simultaneously (e.g., at time step 1) on all TypeA path segments. By example and without limitation, this first set ofCNOT operations comprises: (1) a CNOT targeting q1 with cx1 acting asthe control; (2) a CNOT targeting q2 with cx2 acting as the control; and(3) a CNOT targeting q7 with cx3 used as the control. Here, these threeCNOT operations are performed in a single time step (simultaneously).

Following the operations on Type A path segments, a second set of CNOToperations is next performed simultaneously (e.g., at time step 2) onall Type B segments. This second set of simultaneous CNOT operationscomprises: (1) a CNOT targeting q5 with cx1 acting as the control; (2) aCNOT targeting q3 with cx2 acting as the control; and (3) a CNOTtargeting q6 with cx3 acting as the control.

Following the above operations, a third set of simultaneous CNOToperations is next performed (e.g., at time step 3). This third setincludes: (1) a CNOT targeting q7 with cx1 acting as the control; (2) aCNOT targeting q6 with cx2 acting as the control; and (3) a CNOTtargeting q5 with cx3 acting as the control.

Finally, a fourth and final set of simultaneous CNOT operations isperformed. This fourth set includes (1) a CNOT targeting q3 with cx1acting as the control; (2) a CNOT targeting q7 with cx2 acting as thecontrol; and (3) a CNOT targeting q4 with cx3 acting as the control.

After a set of CNOT operations is performed for each of the G differentedge types, a final measurement step is performed to measure all theancillas (check bits), in which the syndrome is read from the checkbits.

Regardless of the size of the syndrome measurement circuit, the abovesequence of operations guarantees that the syndrome can be measured, ineither the X-basis or the Z-basis, in a constant depth consisting of 2+Gsteps, where G is again the number of edge types (colors in the coloringscheme) as described above. In the above example where G=4, the X-basisor Z-basis syndrome can be read from the entire syndrome measurementcircuit (e.g., from all the check bits) in 6 total measurements steps Incontrast, the traditional methodology described above with respect toFIG. 2 leads to 18 total measurement steps when used to measure thesyndrome of the same graph (the Tanner graph 400).

The above-proposed constant-depth syndrome extraction circuit design isobtained from a pair of sequential syndrome extraction circuits, one forthe X stabilizer generators and one for the Z stabilizer generator.Thus, X stabilizers are measured simultaneously and Z stabilizers aremeasured simultaneously but independent of the X stabilizers. Thiscircuit construction can be applied to any CSS stabilizer code.

Another circuit design disclosed below with respect to FIG. 5-7B adaptsthe above-disclosed edge coloring methodology to further reduce thedepth of the syndrome extraction circuit in quantum devices thatimplement a type of code known as hypergraph product codes (HPGs). Thisadditional reduction in circuit length is achieved via a methodologythat permits simultaneous measurement of the X stabilizers and the Zstabilizers.

FIG. 5 introduces another notation for a Tanner graph 500 (as a modifiedgraph 502) useful in illustrating a methodology for extracting asyndrome from a fully-connected quantum circuit implementing ahypergraph (HPG) product code. Hypergraph product codes are derived bytaking the product of two different linear codes, and are popularbecause they achieve a large minimum distance (which translates to agood error correction performance), are defined by low weight checks(which make it easier to measure the syndrome of these codes), andbecause they offer lower overhead (e.g., many logical qubits can beencoded together in a same data block.

As shown in FIG. 5, the Tanner graph 500 can be visually condensed intoa 1D line of data qubits (circles) and check qubits (squares) as shownin modified graph 502. Linear codes of the form shown by the 1D line inthe modified graph 502 may be used as the X and Y axis of the graphsshown and discussed below with respect to FIG. 6.

FIG. 6 illustrates an exemplary construction of a product graph 600representing an HPG code formed by multiplying together two linear CSScodes, shown along the y-axis 602 and x-axis 604 as “Code 1” and “Code2” respectively. Each of the linear CSS codes consists of multiple checkbits (squares) and data bits (circles), using the same notation shownintroduced in FIG. 5. For simplicity of illustration, only a subset ofthe edges within each graph are shown in the y-axis 602 and the x-axis604. For example, all edge of code one are shown in a firstrepresentation 606 but several are redacted for simplicity in a secondrepresentation shown along the y-axis 602.

The product graph 600 is created by multiplying the two classical linearcodes (Code 1, Code 2) together. Visually, this can be understood bydepicting a Tanner graph for Code 1 along the Y-axis and a Tanner graphfor Code 2 along the X-axis, as shown. A node multiplication step isperformed in which each node on the X-axis is multiplied by each node onthe Y-axis according to a particular product definition methodologyshown in key 610 of FIG. 6, also described as follows:

-   -   a data bit (circle) multiplied by a data bit (circle) yields a        data qubit (circle);    -   a check bit (square) multiplied by another check bit (square)        yields a data qubit (circle);    -   a data bit (circle) multiplied by a check bit (square) yields an        X-basis check qubit (sometimes referred to as a “measure-X        qubit”0) and    -   a check bit (square) multiplied by a data bit (circle) yields a        Z-basis check qubit (sometimes referred to as a “measure-Z        qubit”).

Similar to the multiplicative methodology applied to the check bits anddata qubits, the edges of Code 1 and Code 2 are also multiplied acrosseach row and column of the product graph 600. For example, a verticaledge 612 is added at a like-position in every column of the table, whilea horizontal edge 614 is added at a like-position in every row of thetable. Other edges of the product graph 600 are redacted in FIG. 6 forsimplicity and clarity of illustration.

According to one implementation, the coloring methodology discussedabove with respect to FIG. 4B can be optionally applied to the productgraph 600 to independently measure the X-basis syndrome and the Y-basissyndrome of the HPG code. For example, the z-basis checks and theirassociated edges may be initially ignored while a coloring scheme isapplied to measure the x-basis stabilizers; then, in a similar manner,the x-basis checks and their associated edges are ignored while acoloring scheme is applied to measure the z-basis stabilizers. Whilethis methodology works and yields a shorter syndrome extraction circuitthan traditional approaches, a further improvement can be realized byapplying a cardinal ordering scheme to the edges (e.g., classifying eachedge as north, south, east, or west) and then selectively coloring edgeswithin directional subgroups of the cardinal ordering scheme. Thisapproach, described in detail with respect to FIG. 7, allows thex-stabilizers and the z-stabilizers to be measured simultaneously.

FIG. 7A illustrates an example partial construction of another HPG codeproduct graph 700 that illustrates a method for simultaneously measuringx-stabilizers and z-stabilizers. The basic principles of constructingthe product graph 700 are similar to those of FIG. 6. Two linear CSScodes—Code 1 and Code 2 —are multiplied, as shown in the key 710, toyield a layout of check bits and data bits illustrated in the productgraph 700. For conceptual clarity, the product graph 700 omits amajority of the edges that would, in actuality, exist. Although notshown, each edge in Code 1 is to be multiplied across the product graph700 resulting in a corresponding vertical edge within each column.Likewise, each edge in Code 2 is to be multiplied across the productgraph 700, resulting in a corresponding horizontal edge within each row.For simplicity and illustration of concept, the product graph 700 omitsthe majority of these edges and illustrates only a subset of the totaledges forming connections with a Z-check bit 702.

Measurement of each of the stabilizers in the product graph 700 isdiscussed with respect to FIG. 7B, below, and more particularly—withrespect to a substructure including the X-check bit 702 and itsassociated edges and neighboring data bits.

FIG. 7B illustrates a subgraph structure 704 of the example productgraph 700. This subgraph structure includes the Z-check bit 702, edgesconnected to the Z-check bit 702, and associated neighboring dataqubits. This structure is useful in explaining further operations forperforming simultaneous measurement of x-stabilizers and z-stabilizersin a quantum circuit implementing an HPG code on fully-connected qubits.

Each edge between two nodes on the product graph 700 may be understoodas representing a CNOT or equivalent joint measurement. Thus, accordingto one implementation, measuring the z-stabilizer corresponding toZ-check bit 702 entails the steps of (1) preparing the Z-check bit 702in an initial state; (2) applying the five CNOTs (corresponding to thefive edges); and (3) measuring the Z-check bit 702.

To determine an optimal ordering for such operations with respect to allstabilizer measurements in the product graph 700, a cardinal orderingscheme is used to group edges and prioritize associated qubitoperations. In FIG. 7, the individual edges coupled to the Z-check bit702 bit are labeled based on their cardinal direction within the productgraph 700 relative to the Z-check bit. For example, the Z-check bit 702has one northern edge (“N”), two southern edges (“S1” and “S2”), oneeastern edge (“E”) and one western edge (“W”).

Given a subcircuit consisting of only CNOT gates, we say that thecircuit is cardinally ordered if all of the east CNOTs are appliedfirst, followed by the north CNOTs, then the south CNOTs, and lastly,the west CNOTs. The proposition shown in Table 1, below illustrates thatthis cardinal ordering is sufficient to compute any stabilizer of andHPG graph using a circuit with fully-connected qubits:

TABLE 1 Proposition: Consider a set of m + 1 stabilizer generators s₁,s₂, . . . , s_(m), s_(m+1), for an HGP code with vertices placed at thestandard coordinates. Furthermore, consider a subcircuit consisting ofthe CNOTs used to measure the stabilizers S_(m) = s₁, s₂, . . . , s_(m)collected together in cardinal order, followed by the sequentialapplication of those CNOTs used to measure s_(m+1). Relabeling the timesteps at which those CNOTs used for measuring s_(m+1) are applied suchthat the whole subcircuit is in cardinal order forms an equivalentsubcircuit. Proof. Notice that swapping the order of a pair ofnon-commuting CNOTs whose support intersects on a single qubitintroduces another CNOT between the other two qubits. Suppose, thats_(m+1) is a Z stabilizer generator and that S_(m′) ⊆ S_(m) is thesubset of generators of the X type with support overlapping the supportof s_(m+1). Trivially, the CNOTs in s_(m+1) commute with all the CNOTsin S_(m)\S_(m′). By the hypergraph product construction, the support ofeach stabilizer generator s_(i′)Â ∈ S_(m′) overlaps exactly twice withthe support of s_(m+1) in one of four possible configurations; see FIG.1(a). Depending on the configuration, we need to perform 0 or 2 swaps ofthe CNOTs in s_(m+1) with non-commuting CNOTs in s_(i′) to bring theminto cardinal order. Each such swap introduces a CNOT gate from thestabilizer qubit of s_(i′) to the stabilizer qubit of s_(m+1). As theseintroduced gates have order two, and commute with all other gates inthe, we can reorder them such that they act consecutively and cancel oneanother. Thus, when reordering S_(m) ∪ s_(m+1), the action of thecircuit is preserved. A similar argument applies when s_(m+1) is an Xtype stabilizer generator.

The proof set forth in Table 1, above, leads to the followingtransformation, referred to herein as Circuit Transformation 2, which isperiodically referred to elsewhere throughout this disclosure:

-   -   Circuit Transformation 2: Given a sequential syndrome extraction        circuit (SSEC) for a HPG code with vertices placed at standard        coordinates (e.g., coordinates corresponding to counterparts in        the connectivity graph for the given circuit), an equivalent        circuit is obtained by permuting the order of the CNOTs such        that they are in cardinal order, as defined above.

Per this methodology, any individual z-stabilizer and x-stabilizer inthe product graph of FIG. 7A can be measured using a subcircuit of CNOTswhere the CNOTs gates are applied in the cardinal ordering scheme: east,north, south, and west.

The below-described methodology expands on this concept further,allowing the z-stabilizers and the z-stabilizers to be measuredsimultaneously by assigning a coloring scheme sub-circuit to eachdifferent cardinal direction within the above-describedcardinally-ordered CNOT circuit. Stated differently, a coloring scheme(similar to that of FIG. 4A-4B) may be applied to edges of a particularcardinal direction when more than one such edge of a given directionexists for any check bit on the product graph 700. An example sequenceof operations for implementing this methodology is enumerated below:

-   -   1. Prepare all x-basis and z-basis check qubits in the        appropriate state during the 0.    -   2. Partition all edges of the associated HPG code into four        subsets: the east, north, south, and west edges (as generally        explained and simplified above with respect to a single check        bit in FIG. 7B).    -   3. If there exist two or more edges of a given cardinal        direction connected any one of the check bits in the HPG code,        assign a coloration scheme to edges of that direction. In the        example of FIG. 7B where there exist two south-direction edges        S1, S2, the edges S1 and S2 are therefore assigned to different        colors (edge types) in a coloring scheme. For example, S1 is a        Type 1 south-direction edge and S2 is a Type 2 south-direction        edge. Likewise, if there exist multiple edges of the other        cardinal direction (N, E, W), those same-direction edges may        also be assigned a color scheme. For example, there may exist a        Type 1 East-direction edge, a Type 2 east-direction edge, etc.    -   4. Rearrange the CNOTs in cardinal order by recursive        applications of Circuit Transformation 2: Loop through each        direction in the cardinal ordering “east, north, south, west,”        one by one. For each selected direction in this loop, perform        step 5, below.    -   5. Compress the subcircuit of each cardinal direction using        Circuit Transformation 1: For each selected direction “D” (N, S,        E, W), loop over the colors “C” assigned to one or more edges of        the selected direction D, simultaneously applying all CNOTs of        color “C” for the selected direction across the entire product        graph. For example, a single time-step may entail applying a        CNOT to all Type 1 South edges (S1) in the entire product graph        700, then—at a next time-step, applying a CNOT to all Type 2        South edges (S2) in the entire product graph 700.    -   6. Measure all check qubits in the appropriate basis during the        final timestep {τ}.

Note, operations (4) and (5) above collectively provide for (1)beginning the CNOTs for each cardinal direction at a different time stepand (2) if a coloring scheme exists for a given cardinal direction,performing the CNOTs for the different colors of that cardinal directionat different time-steps.

In other implementations, a subcircuit providing the functionality ofthe circuit described above (with respect to steps 1-6) includes one ormore alternative 2-qubit gates in lieu of each CNOT. For example, thefunctionality of the “CNOT” gate could be implemented by a control-Zgate or a joint measurement.

In addition to the reductions in depth of the syndrome extractioncircuit realized above, still further improvements in depth can berealized for HPG codes by a technique that provides for drawing theproduct graph for the HPG in according to a construct that seeks tobalance edges of each cardinal direction. Specifically, this methodologyprovides for algorithmic manipulations to the HPG product code graph fora given HPG circuit implementation to ensure that the number of northand south edges extending from each check bit is balanced and to ensurethat the number of east and west edges extending from each check bit isbalanced.

As used herein, vertical edges are said to be “balanced” for a given bitwhen the number of north edges extending from the bit is equal to thenumber of south edges extending from the bit, within +/−1 edge (e.g.,the remainder occurring when the number of vertical edges is odd);likewise, horizontal edges are said to be “balanced” for a given bitwhen the number of east edges extending from the bit is equal to thenumber of west edges extending from the bit, within +/−1 edge (e.g., theremainder occurring when the number of horizontal edges is odd).

FIG. 8 illustrates an example methodology for re-writing a CSS linearcode 800 according to a balanced cardinal ordering scheme that permits afurther reduction in depth of an associated syndrome extraction circuit.

Referring to View A of FIG. 8, an exemplary CSS linear code 800 isshown. This CSS linear code may, for example, form either of themultipliers (Code 1, Code 2) used to form the HPG codes illustrated withrespect to FIGS. 6 and 7. For illustrative purposes, the data bits andcheck bits of the CSS linear code 800 are shown labeled numerically fromtop to bottom (1-7). By example and without limitation, the CSS linearcode 800 includes two check bits (bit number 2 and number 6) and fivedata bits (bit numbers 1, 3, 4, 5, and 7). Here, the first check bit(number 2) has four vertical edges, including one north edge extendingto bit number 1 and three south edges extending to bit numbers 3, 4, and5, respectively. Likewise, the second check bit (number 6) has threevertical edges, including two north edges extending to bit numbers 4 and5 and a south edge extending to bit number 7. The representation of theCSS linear code 800 shown in View A is considered “unbalanced” due tothe 1-to-3 ratio of north to south edges on bit number 2.

View B illustrates an alternate depiction of the CSS linear code 800 inwhich the bits are arranged, in numerical order, around thecircumference of a circle. Edges shown in View A are preserved. Sincethe CSS linear code 800 is presumed to be implemented in a quantumcircuit with fully connected qubits, the relative ordering of the bitsalong this circle can be altered without affecting the code. When anaxis 804 is drawn through bit number 2, it can be seen that the edgesare “unbalanced” due to the fact that this bit has three edges on oneside of the axis 804 and one edge on the other side of the axis 804.

View C illustrates a functionally equivalent but visually differentrepresentation of the CSS linear code 800. Between Views B and C, thebits of have been reordered along the circumference of the circle whilepreserving the edges between each numbered pair of bits. In thisalternate representation, an axis 806 drawn through bit number 2 dividesthe circle in half and thereby illustrates that the four edges of bitnumber 2 (e.g., extending to the same bits as in Views A and B) are nowbalanced in the sense that bit number 2 is connected to a same number ofedges on each side of the axis 806.

Still referring to the representation of View C, the second check bit(number 6) is also said to be balanced because there exist an odd numberof edges that are as balanced as possible (e.g., equal +/−1 edge) onopposite sides of an axis 808 intersecting this bit and dividing thecircle in half.

View D illustrates yet another functionally equivalent representation ofthe CSS linear code 800 that relies on the same bit ordering scheme asthat shown in View C. That is, the numerical bits remain arrangedrelative to one another in a manner that matches that of the bitarrangement along the circle in view C. This representation of the CSSlinear code 800 is said to be “balanced.”

In sum, although Views A and D illustrate functionally equivalentrepresentations of the CSS linear code 800, the representation of View Aprovides for an unbalanced cardinal ordering while the representation ofView D provides for a balanced cardinal ordering. If the View Drepresentation of the CSS linear code 800 is used generate the productcode graph used defining the syndrome extraction circuit (e.g., in amanner the same or similar to that described with respect to FIGS. 6 and7), the resulting product code graph then has a balanced number of northv. south vertical edges for each check bit and a balanced number ofeast. V. west horizontal edges.

When the stabilizer circuit described above with respect to FIG. 7A and7B is applied to a product graph with a balanced cardinal ordering edgescheme (e.g., generated based on the “View D” representation FIG. 8),the resulting syndrome extraction circuit is of a shorter depth thanthat employed to extract the syndrome from a product graph generatedbased on the unbalanced “View A” representation of the same linear code.The aforementioned reduction in depth of the stabilizer circuit for anHPG codes generated based on a balanced cardinal ordering (as describedabove) is further supported by the proof in Table 2 below:

TABLE 2 Proposition: Consider an HPG code of length N with M stabilizergenerators and with vertices placed at the standard coordinates. Ifδ_(E), δ_(N), δ_(S) and δ_(W) are respectively the maximum degrees ofthe east, north, south and west subgraphs, then Circuit construction 1uses N + M physical qubits and has depth δ_(E) + δ_(N) + δ_(S) +Âδ_(W) + 2. Proof. A data qubit is assigned to each qubit vertex of theTanner graph and a readout qubit is assigned to each stabilizer vertexof the Tanner graph leading to a total of N + M physical qubits.Furthermore, the adjacency graph G(C_(E)) of the east subcircuit C_(E)is isomorphic to the east subgraph of the Tanner graph. Thus, G(C_(E))has chromatic number δ_(E). The same holds for the three otherdirections. Two additional time steps are required to apply thenecessary state preparations and measurements leading to a depth ofδ_(E) + δ_(N) + δ_(S) + δ_(W) + 2.

The proposition and the accompanying proof in Table 2, above ,establishes that the number of time-steps in the stabilizer circuit fora fully connected HPC code is given by the relation δ+2 where δ is thesum of the number of edges in each of the four cardinal directions inthe associated product code graph (e.g., δ=δ_(E)+δ_(N)+δ_(S)+δ_(W)).

According to one implementation, algorithmic processing is utilized toidentify a balanced cardinal edge ordering scheme for any linear codeprior to generation of the product code graph and/or to theimplementation of the product code in a quantum architecture.

FIG. 9 illustrates example operations 900 for measuring a syndrome in aquantum circuit with fully connected qubits implementing a CSSstabilizer code. A graph generation operation 902 generates a Tannergraph representing the CSS code implemented and includes check nodescorresponding to the syndrome qubits (also referred to herein as checkqubits) of the quantum circuit and data nodes corresponding to dataqubits of the circuit.

A selection operation 904 selects either the X-basis or the Z-basis forfault measurement. A defining operation 906 defines an integer “G” asthe maximum number of edges abutting any one of the check nodes of theselected basis and defines G-number of different edge types. Thedefining operation 906 further defines G number of different edge typesto be assigned to the edges of the Tanner graph.

For each of the check nodes of the selected basis, an assignmentoperation 908 assigns the different edges coupled to the check node to aselect one of the G-number of different edge types. This assignment ofedges to edge types is performed in accordance with a constraintrequiring that no two of the edges abutting a same one of the checknodes is assigned to a same one of the G different edge types.

A preparation operation 910 prepares the syndrome bits of the selectedbasis (e.g., X-basis syndrome bits or Z-basis syndrome bits) in a knownstate. The preparation operation 910 is, for example, performed by acontroller than generates and transmits a control signal to the quantumcircuit.

A qubit manipulation operations 912 performs G-number oftemporally-separated rounds of multi-qubit operations. Each of theG-number of temporally-separated rounds is associated with a differentone of the G-number of edge types, and the set of multi-qubit operationsperformed during each of the temporally-separated round entangles ofsets of qubits connected to edges assigned to the edge type associatedwith that round.

A syndrome extraction operation 914 extract a syndrome for the selectedbasis by measuring the syndrome bits of that basis (e.g., measuring allX-basis syndrome bits). A determination operation 916 determines whetheran alternate fault basis remains to be measured to complete the syndromemeasurement. If, for example, the X-basis is measured first, thesyndrome measurement next entails similar measurements in the Z-basis.In this case, a selection operation 918 selects the alternate faultbasis as the new basis for fault measurement and the operations 908-916are repeated with respect to the newly-selected basis for faultmeasurement.

According to one implementation, the operations 900 provide forextraction of the X-basis syndrome in 2+G measurement rounds andextraction of the Z-basis syndrome in another 2+G measurement rounds.This result holds true regardless of the CSS code implemented by thequantum architecture.

FIG. 10 illustrates example operations 1000 for measuring a syndrome ina quantum circuit with fully connected qubits implementing an HPG codethat is formed by multiplying two linear CSS codes together. Anidentification operation 1002 identifies the two linear code multipliersof the HPG code.

A cardinal ordering operation 1004 identifies, for each of the twolinear CSS codes, a corresponding Tanner graph representation o the codethat implements a balanced cardinal ordering edge scheme. The balancedcardinal ordering edge scheme is one that provides a balanced number ofedges extending in opposing directions from each of the check nodes inthe Tanner graph, as generally defined and discussed above with respectto FIG. 8.

A product code graph generation operation 1006 generates a product codefor the HPG by multiplying together the two Tanner graphs identified bythe cardinal ordering operation. This linear code multiplication is, forexample, achieved by performing operations consistent with those shownand described with respect to FIG. 6.

A cardinal direction assignment operation 1008 assigns a cardinaldirection (e.g., N, S, E, W) to each edge in the product code graph suchthat the assigned cardinal direction indicates an orientation of theedge relative to a check node forming an endpoint of the edge.

A selection operation 1010 selects one of the cardinal directions. As isexplained below, the operations 1010 is looped through four times—onetime with respect to each of the four cardinal directions. According toone implementation, the selection of such cardinal operations isperformed in the order: East, North, South, West (note: this ordering isconsistent with the proposition and proof provided above in Table 1).

An identification and assignment operation 1012 performs assignmentoperations with respect to each check node in the product code graph.Specifically, for each check node in the product code graph, theidentification and assignment operation 1012 (1) identifies a subset ofedges coupled to the check node that are assigned to thecurrently-selected cardinal direction; and (2) assigns a different oneof up to G-number of edge types to each edge within the identifiedsubset of edges. At the conclusion of the identification and assignmentoperation 1012, all edges in the product code graph of thecurrently-selected cardinal direction are assigned to one of theG-number of edge types. Here, “G” may be defined as the maximum numberof edges abutting any check node in the product code graph.

A determination operation 1014 determines whether any of the fourcardinal directions remains to be selected by selection operation 1010.If so, the loop comprising 1010, 1012 is repeated until all fourcardinal directions have been selected and edges associated with suchdirections have each been assigned to one of the G-number of differentedge types.

A syndrome bit preparation operation 1016 next prepares all of thesyndrome bits in a known state. A qubit manipulation operation 1018conducts multiple time-separated rounds of multi-qubit operations, eachround including operations to concurrently entangle select pairs ofqubits (e.g., CNOT operations), where each pair being entangled in thesame round is connected to an edge assigned to a same cardinal directionand a same edge type for the cardinal direction. Stated differently, thequbit manipulation operation 1018 performs a number of time-separatedrounds of concurrent entanglement operations that equals a sum of δ_(E)(the number of different edge types assigned to east-direction edges),δ_(N) (the number of different edge types assigned to north-directionedges), δ_(S), (the number of different edge types assigned tosouth-direction edges), and δ_(W) (the number of different edge typesassigned to west-direction edges).

A syndrome extraction operation 1020 extracts the syndrome from thequantum circuit by simultaneously measuring all of the syndrome bits inboth the X-basis and the X-basis.

According to one implementation, the operations 900 provide forextraction of the entire syndrome (in both X-basis and X-basis) in 2+6measurement rounds, wherein δ=δ_(N)+δ_(S)+δ_(E)+δ_(W). This constantdepth syndrome extraction is achievable regardless of the linear codesused to generate the HPG code .

FIG. 11 illustrates an exemplary computing environment 1100 suitable forimplementing aspects of the disclosed technology. This figure and thefollowing discussion are intended to provide a brief, generaldescription of an exemplary computing environment in which the disclosedtechnology may be implemented. Although not required, the disclosedtechnology is described in the general context of computer executableinstructions, such as program modules, being executed by a personalcomputer (PC). Generally, program modules include routines, programs,objects, components, data structures, etc., that perform particulartasks or implement particular abstract data types. Moreover, thedisclosed technology may be implemented with other computer systemconfigurations, including hand held devices, multiprocessor systems,microprocessor-based or programmable consumer electronics, network PCs,minicomputers, mainframe computers, and the like. The disclosedtechnology may also be practiced in distributed computing environmentswhere tasks are performed by remote processing devices that are linkedthrough a communications network. In a distributed computingenvironment, program modules may be located in both local and remotememory storage devices.

The computing environment 1100 of FIG. 11 includes a classical computingenvironment 1101 (e.g., a PC) coupled to one or more quantum computedevice(s) 1114. The quantum compute devices 1114 includes at least aquantum register 1120 and a readout device 1122 including quantumhardware implementing measurement circuits 1124. Aspects of the quantumcompute devices 1114 may be controlled by hardware and software elementsof the classical computing system 1101.

In one implementation, the classical computing system 1101 includes oneor more processing units 1102, a system memory 1104, and a system bus1106 that couples various system components including the system memory1104 to the one or more processing units 1102. The system bus 1106 maybe any of several types of bus structures including a memory bus ormemory controller, a peripheral bus, and a local bus using any of avariety of bus architectures. The exemplary system memory 1104 includesread only memory (ROM) 1108 and random access memory (RAM) 1110. A basicinput/output system (BIOS) 1112, containing the basic routines that helpwith the transfer of information between elements within the classicalcomputing system 1101 is stored in the ROM 1108.

In one implementation, the system memory 1104 stores short-depthsyndrome extraction logic 120 of FIG. 1 and also stores decoding logicfor detecting and correcting errors in measurement data using syndromesextracted by executing the short-depth syndrome extraction logic.

The classical computing system 1101 further includes one or more storagedevices 1130 such as a hard disk drive for reading from and writing to ahard disk, a magnetic disk drive for reading from or writing to aremovable magnetic disk, and an optical disk drive for reading from orwriting to a removable optical disk (such as a CD-ROM or other opticalmedia). Such storage devices can be connected to the system bus 1106 bya hard disk drive interface, a magnetic disk drive interface, and anoptical drive interface, respectively. The drives and their associatedcomputer readable media provide nonvolatile storage of computer-readableinstructions, data structures, program modules, and other data for theclassical computing system 1101. Other types of computer-readable mediawhich can store data that is accessible by a PC, such as magneticcassettes, flash memory cards, digital video disks, CDs, DVDs, RAMs,ROMs, and the like, may also be used in the exemplary operatingenvironment.

A number of program modules may be stored in the storage devices 1130including an operating system, one or more application programs, otherprogram modules, and program data. A user may enter commands andinformation into the classical computing system 1101 through one or moreinput devices 1140 such as a keyboard and a pointing device such as amouse. Other input devices may include a digital camera, microphone,joystick, game pad, satellite dish, scanner, or the like. These andother input devices are often connected to the one or more processingunits 1102 through a serial port interface that is coupled to the systembus 1106, but may be connected by other interfaces such as a parallelport, game port, or universal serial bus (USB). A monitor 1146 or othertype of display device is also connected to the system bus 1106 via aninterface, such as a video adapter. Other peripheral output devices1145, such as speakers and printers (not shown), may be included.

The classical computing system 1101 may operate in a networkedenvironment using logical connections to one or more remote computers,such as a remote computer 1160. In some examples, one or more network orcommunication connections 1150 are included. The remote computer 1160may be another PC, a server, a router, a network PC, or a peer device orother common network node, and typically includes many or all of theelements described above, although only a memory storage device 1162 hasbeen illustrated in FIG. 11. The classical computing system 1101 and/orthe remote computer 1160 can be connected to a logical a local areanetwork (LAN) and a wide area network (WAN). Such networkingenvironments are commonplace in offices, enterprise wide computernetworks, intranets, and the Internet.

When used in a LAN networking environment, the classical computingsystem 1101 is connected to the LAN through a network interface. Whenused in a WAN networking environment, the classical computing system1101 typically includes a modem or other means for establishingcommunications over the WAN, such as the Internet. In a networkedenvironment, program modules depicted relative to the classicalcomputing system 1101, or portions thereof, may be stored in the remotememory storage device or other locations on the LAN or WAN. The networkconnections shown are exemplary, and other means of establishing acommunications link between the computers may be used.

The classical computing system 1101 may include a variety of tangiblecomputer-readable storage media and intangible computer-readablecommunication signals. Tangible computer-readable storage can beembodied by any available media that can be accessed by the classicalcomputing system 1101 and includes both volatile and nonvolatile storagemedia, removable and non-removable storage media. Tangiblecomputer-readable storage media excludes intangible and transitorycommunications signals and includes volatile and nonvolatile, removableand non-removable storage media implemented in any method or technologyfor storage of information such as computer readable instructions, datastructures, program modules or other data. Tangible computer-readablestorage media includes, but is not limited to, RAM, ROM, EEPROM, flashmemory or other memory technology, CDROM, digital versatile disks (DVD)or other optical disk storage, magnetic cassettes, magnetic tape,magnetic disk storage or other magnetic storage devices, or any othertangible medium which can be used to store the desired information, andwhich can be accessed by the classical computing system 1101. Incontrast to tangible computer-readable storage media, intangiblecomputer-readable communication signals may embody computer readableinstructions, data structures, program modules or other data resident ina modulated data signal, such as a carrier wave or other signaltransport mechanism. The term “modulated data signal” means a signalthat has one or more of its characteristics set or changed in such amanner as to encode information in the signal. By way of example, andnot limitation, intangible communication signals include wired mediasuch as a wired network or direct-wired connection, and wireless mediasuch as acoustic, RF, infrared and other wireless media.

An example method for extracting a syndrome from a quantum measurementcircuit includes generating a graph representing a code implemented bythe quantum measurement circuit. The graph includes bit nodescorresponding to the data qubits in the quantum measurement circuit,check nodes corresponding to the syndrome qubits in the quantummeasurement circuit, and edges between the bit nodes and check nodeseach being associated with a stabilizer measurement provided by thecode. The method further provides for assigning each of the edges in thegraph to a select one of G-number of different edge types and performingat least G-number of temporally-separated rounds of multi-qubitoperations, each of the temporally-separated rounds of multi-qubitoperations enacting concurrent multi-qubit operations on endpoints of asubset of the edges assigned to a same one of the G different edgetypes.

In yet another example method of any preceding method, the methodprovides for concurrent measurement of multiple stabilizers.

In still another example method of any preceding method, assigning eachof the edges in the graph to a select one of the G number of thedifferent edge types further comprises ensuring that no two of the edgesabutting a same one of the check nodes or a same one of the qubit nodesis assigned to a same one of the G different edge types, the number Grepresenting a maximum number of edges abutting any individual one ofthe check nodes and bit nodes in the graph.

In still another example method of any preceding method, the code is ahypergraph product code and assigning each of the edges in the graph toa select one of the G number of the different edge types furthercomprises assigning a cardinal direction to each edge in the graph basedon an orientation of the edge within the graph relative to a check nodecoupled to the edge. For each one of the check nodes, the method furtherprovides for identifying one or more same-directional subsets of edgescoupled to the node, each one of the same-directional subsets of theedges including edges coupled to the check node and assigned to a sameone of the cardinal directions. A different one of the G number of edgetypes is then assigned to each edge within each one of the identifiedsame-directional subsets of edges coupled to each of the check nodes.

In another example method of any preceding method, performing the atleast G-number of temporally-separated rounds of qubit operationsfurther comprises conducting multiple time-separated rounds ofmulti-qubit operations, and simultaneously measuring the syndrome bitsin the quantum measurement circuit to extract the syndromesimultaneously with respect to both x-basis check qubit and z-basischeck qubits. Each round of the multi-qubit operations includesoperations to concurrently entangle select pairs of qubits connected toedges assigned to a same cardinal direction and a same edge type.

In yet still another example method of any preceding method, generatingthe graph further comprises balancing a number of the edges coupled toeach individual one of the check nodes that are associated with northand south directions of the cardinal directions, and balancing a numberof the edges coupled to each individual one of the check nodes that areassociated with east and west directions of the cardinal directions.

In another example method of any preceding method, the concurrentmulti-qubit operations performed during each of the temporally-separatedrounds include multiple CNOT operations, each of the CNOT operationswithin a same one of the temporally-separated rounds being effective toentangle pairs of qubits connected via a same edge type in the graph.

In yet another example method of any preceding method, performing the atleast G-number of temporally-separated rounds of qubit operation furthercomprises: preparing the syndrome bits in a known state and performing Gnumber of temporally-separated rounds of multi-qubit operations. Themulti-qubit operations performed during each of the rounds beingeffective to entangle sets of qubits connected to edges that share acommon edge type of the G different edge types. Finally, the methodprovides for extracting the syndrome by measuring the syndrome bits.

In still another example method of any preceding method, the qubits arefully connected in the quantum measurement circuit.

An example quantum system includes a quantum measurement circuit withdata qubits and syndrome qubits. The system includes a means forgenerating a graph representing a code implemented by the quantummeasurement circuit. The graph includes bit nodes corresponding to dataqubits of the quantum measurement circuit, check nodes corresponding tothe syndrome qubits of the quantum measurement circuit, and edgesbetween the bit nodes and check nodes each being associated with astabilizer measurement provided by the code. The system further includesa means for assigning each of the edges in the graph to a select one ofG-number of different edge types, and a means for performing at leastG-number of temporally-separated rounds of multi-qubit operations, eachof the temporally-separated rounds of multi-qubit operations enactingconcurrent multi-qubit operations on endpoints of a subset of the edgesassigned to a same one of the G different edge types.

An example system disclosed herein includes a controller and a quantummeasurement circuit implementing an error correction code and includingdata qubits and syndrome qubits. The controller is configured togenerate a graph representing a code implemented by the quantummeasurement circuit, where the graph includes bit nodes corresponding tothe data qubits, check nodes corresponding to the syndrome qubits, andedges between the bit nodes and check nodes each being associated with astabilizer measurement provided by the code. The controller is furtherconfigured to assign each of the edges in the graph to a select one of“G” number of different edge types and to extract a syndrome from thequantum measurement circuit via a methodology that includes performingat least G-number of temporally-separated rounds of multi-qubitoperations, each of the temporally-separated rounds of multi-qubitoperations enacting concurrent multi-qubit operations on endpoints of asubset of the edges assigned to a same one of the G different edgetypes.

In another example system of any preceding system, extracting thesyndrome includes concurrently measuring multiple stabilizers of theerror correction code.

In yet still another example system of any preceding system, assigningeach of the edges in the graph to a select one of the G number of thedifferent edge types further comprises ensuring that no two of the edgesabutting a same one of the check nodes is assigned to a same one of theG different edge types, where the number G represents a maximum numberof edges abutting any individual one of the check nodes and bit nodes inthe graph.

In still another example system of any preceding system, the code is ahypergraph product code and assigning each of the edges in the graph toa select one of the G number of the different edge types furthercomprises assigning a cardinal direction to each edge in the graph basedon an orientation of the edge within the graph relative to a check nodecoupled to the edge. For each one of the check nodes, the controller isconfigured to identify one or more same-directional subsets of edgescoupled to the node, each of the same-directional subsets of the edgesincluding edges coupled to the check node and assigned to a same one ofthe cardinal directions. The controller assigns a different one of the Gnumber of edge types to each edge within each one of the identifiedsame-directional subsets of edges coupled to each of the check nodes.

In another example system of any preceding system, performing the atleast G-number of temporally-separated rounds of multi-qubit operationsfurther comprises conducting one or more time-separated rounds ofmulti-qubit operations and simultaneously measuring the syndrome bits inthe quantum measurement circuit to extract the syndrome simultaneouslywith respect to both x-basis check qubit and z-basis check qubits. Eachround of the multi-qubit operations includes operations to concurrentlyentangle select pairs of qubits connected to edges assigned to a samecardinal direction and a same edge type.

In yet another example system of any preceding system, generating thegraph further comprises balancing a number of the edges coupled to eachindividual one of the check nodes that are associated with north andsouth directions of the cardinal directions and balancing a number ofthe edges coupled to each individual one of the check nodes that areassociated with east and west directions of the cardinal directions.

In yet still another example system of any preceding system, theconcurrent multi-qubit operations performed during each of thetemporally-separated rounds include multiple CNOT operations, each ofthe CNOT operations within a same one of the temporally-separated roundsbeing effective to entangle pairs of qubits connected via a same edgetype in the graph.

In still another example system of any preceding system, performing theat least G-number of temporally-separated rounds of multi-qubitoperations further comprises preparing the syndrome bits in a knownstate and performing the G number of temporally-separated rounds ofmulti-qubit operations after preparing the syndrome bits in the knownstate. The multi-qubit operations performed during each of the rounds iseffective to entangle sets of qubits connected to edges that share acommon edge type of the G different edge types. The controller isconfigured to extract the syndrome from the quantum measurement circuitby measuring the syndrome bits.

In still yet another example system of any preceding system, the qubitsare fully connected in the quantum measurement circuit.

An example computer-readable storage media disclosed herein encodescomputer-executable instructions for executing a computer process toextract a syndrome from a quantum measurement circuit. The computerprocess comprises generating a graph representing a code implemented bythe quantum measurement circuit, the graph including bit nodescorresponding to data qubits in the quantum measurement circuit, checknodes corresponding to syndrome qubits in the quantum measurementcircuit, and edges between the bit nodes and check nodes each beingassociated with a stabilizer measurement provided by the code. Thecomputer process further comprises assigning each of the edges in thegraph to a select one of G number of different edge types and performingat least G-number of temporally-separated rounds of qubit operations.Each of the temporally-separated rounds of qubit operations enactsconcurrent multi-qubit operations on endpoints of a subset of the edgesassigned to a same one of the G different edge types.

In another example tangible computer-readable storage media disclosedherein, the encoded computer process provides for concurrent measurementof multiple stabilizers.

The implementations described herein are implemented as logical steps inone or more computer systems. The logical operations may be implemented(1) as a sequence of processor-implemented steps executing in one ormore computer systems and (2) as interconnected machine or circuitmodules within one or more computer systems. The implementation is amatter of choice, dependent on the performance requirements of thecomputer system being utilized. Accordingly, the logical operationsmaking up the implementations described herein are referred to variouslyas operations, steps, objects, or modules. Furthermore, it should beunderstood that logical operations may be performed in any order, unlessexplicitly claimed otherwise or a specific order is inherentlynecessitated by the claim language. The above specification, examples,and data, together with the attached appendix, provide a completedescription of the structure and use of exemplary implementations.

The above specification, examples, together with the attached appendixprovide a complete description of the structure and use of exemplaryimplementations. Since many implementations can be made withoutdeparting from the spirit and scope of the claimed invention, the claimshereinafter appended define the invention. Furthermore, structuralfeatures of the different examples may be combined in yet anotherimplementation without departing from the recited claims. The abovespecification, examples, and data provide a complete description of thestructure and use of exemplary implementations. Since manyimplementations can be made without departing from the spirit and scopeof the claimed invention, the claims hereinafter appended define theinvention. Furthermore, structural features of the different examplesmay be combined in yet another implementation without departing from therecited claims.

What is claimed is:
 1. A method for extracting a syndrome from a quantummeasurement circuit, the quantum measurement circuit including dataqubits and syndrome qubits; generating a graph representing a codeimplemented by the quantum measurement circuit, the graph including bitnodes corresponding to the data qubits, check nodes corresponding to thesyndrome qubits, and edges between the bit nodes and check nodes eachbeing associated with a stabilizer measurement provided by the code;assigning each of the edges in the graph to a select one of G-number ofdifferent edge types; and performing at least G-number oftemporally-separated rounds of multi-qubit operations, each of thetemporally-separated rounds of multi-qubit operations enactingconcurrent multi-qubit operations on endpoints of a subset of the edgesassigned to a same one of the G different edge types.
 2. The method ofclaim 1, wherein the method provides for concurrent measurement ofmultiple stabilizers.
 3. The method of claim 1, wherein assigning eachof the edges in the graph to a select one of the G-number of thedifferent edge types further comprises: ensuring that no two of theedges abutting a same one of the check nodes or a same one of the qubitnodes is assigned to a same one of the G-number of different edge types,the number G representing a maximum number of edges abutting anyindividual one of the check nodes and bit nodes in the graph.
 4. Themethod of claim 1,wherein the code is a hypergraph product code andassigning each of the edges in the graph to a select one of the G-numberof the different edge types further comprises: assigning a cardinaldirection to each edge in the graph based on an orientation of the edgewithin the graph relative to a check node coupled to the edge; for eachone of the check nodes, identifying one or more same-directional subsetsof edges coupled to the node, each same-directional subset of the edgesincluding edges coupled to the check node and assigned to a same one ofthe cardinal directions; assigning a different one of the G-number ofedge types to each edge within each one of the identifiedsame-directional subsets of edges coupled to each of the check nodes. 5.The method of claim 4, wherein performing the at least G-number oftemporally-separated rounds of qubit operations further comprises:conducting multiple time-separated rounds of multi-qubit operations,each round including operations to concurrently entangle select pairs ofqubits connected to edges assigned to a same cardinal direction and asame edge type; simultaneously measuring the syndrome bits in thequantum measurement circuit to extract the syndrome simultaneously withrespect to both x-basis check qubit and z-basis check qubits.
 6. Themethod of claim 4, wherein generating the graph further comprises:balancing a number of the edges coupled to each individual one of thecheck nodes that are associated with north and south directions of thecardinal directions; and balancing a number of the edges coupled to eachindividual one of the check nodes that are associated with east and westdirections of the cardinal directions.
 7. The method of claim 1, whereinthe concurrent multi-qubit operations performed during each of thetemporally-separated rounds include multiple CNOT operations, each ofthe CNOT operations within a same one of the temporally-separated roundsbeing effective to entangle pairs of qubits connected via a same edgetype in the graph.
 8. The method of claim 1, wherein performing the atleast G-number of temporally-separated rounds of qubit operation furthercomprises: preparing the syndrome bits in a known state; after preparingthe syndrome bits in the known state, performing the G-number oftemporally-separated rounds of multi-qubit operations, the multi-qubitoperations performed during each of the rounds being effective toentangle sets of qubits connected to edges that share a common edge typeof the G different edge types; and extracting the syndrome by measuringthe syndrome bits.
 9. The method of claim 1, wherein the qubits arefully connected in the quantum measurement circuit.
 10. A systemcomprising: a quantum measurement circuit implementing an errorcorrection code and including data qubits and syndrome qubits; acontroller configured to: generate a graph representing a codeimplemented by the quantum measurement circuit, the graph including bitnodes corresponding to the data qubits, check nodes corresponding to thesyndrome qubits, and edges between the bit nodes and check nodes eachbeing associated with a stabilizer measurement provided by the code;assign each of the edges in the graph to a select one of “G” number ofdifferent edge types; and extracting a syndrome from the quantummeasurement circuit via a methodology that includes performing at leastthe G-number of temporally-separated rounds of multi-qubit operations,each of the temporally-separated rounds of multi-qubit operationsenacting concurrent multi-qubit operations on endpoints of a subset ofthe edges assigned to a same one of the G-number of different edgetypes.
 11. The system of claim 10, wherein extracting the syndromeincludes concurrently measuring multiple stabilizers of the errorcorrection code.
 12. The system of claim 10, wherein assigning each ofthe edges in the graph to a select one of the G-number of the differentedge types further comprises: ensuring that no two of the edges abuttinga same one of the check nodes is assigned to a same one of the G-numberof different edge types, the number G representing a maximum number ofedges abutting any individual one of the check nodes and bit nodes inthe graph.
 13. The system of claim 10,wherein the code is a hypergraphproduct code and assigning each of the edges in the graph to a selectone of the G-number of the different edge types further comprises:assigning a cardinal direction to each edge in the graph based on anorientation of the edge within the graph relative to a check nodecoupled to the edge; for each one of the check nodes, identifying one ormore same-directional subsets of edges coupled to the node, eachsame-directional subset of the edges including edges coupled to thecheck node and assigned to a same one of the cardinal directions;assigning a different one of the G-number of edge types to each edgewithin each one of the identified same-directional subsets of edgescoupled to each of the check nodes.
 14. The system of claim 13, whereinperforming the at least G-number of temporally-separated rounds ofmulti-qubit operations further comprises: conducting one or moretime-separated rounds of multi-qubit operations, each round includingoperations to concurrently entangle select pairs of qubits connected toedges assigned to a same cardinal direction and a same edge type;simultaneously measuring the syndrome bits in the quantum measurementcircuit to extract the syndrome simultaneously with respect to bothx-basis check qubit and z-basis check qubits.
 15. The system of claim13, wherein generating the graph further comprises: balancing a numberof the edges coupled to each individual one of the check nodes that areassociated with north and south directions of the cardinal directions;and balancing a number of the edges coupled to each individual one ofthe check nodes that are associated with east and west directions of thecardinal directions.
 16. The system of claim 10, wherein the concurrentmulti-qubit operations performed during each of the temporally-separatedrounds include multiple CNOT operations, each of the CNOT operationswithin a same one of the temporally-separated rounds being effective toentangle pairs of qubits connected via a same edge type in the graph.17. The system of claim 10, wherein performing the at least G-number oftemporally-separated rounds of multi-qubit operations further comprises:preparing the syndrome bits in a known state; after preparing thesyndrome bits in the known state, performing the G number oftemporally-separated rounds of multi-qubit operations, the multi-qubitoperations performed during each of the rounds being effective toentangle sets of qubits connected to edges that share a common edge typeof the G-number of different edge types; and extracting the syndrome bymeasuring the syndrome bits.
 18. The system of claim 10, wherein thequbits are fully connected in the quantum measurement circuit.
 19. Oneor more tangible computer-readable storage media encodingprocessor-executable instructions for executing a computer process toextract a syndrome from a quantum measurement circuit, the quantummeasurement circuit including data qubits and syndrome qubits, thecomputer process comprising: responsive to the instruction, generating agraph representing a code implemented by the quantum measurementcircuit, the graph including bit nodes corresponding to the data qubits,check nodes corresponding to the syndrome qubits, and edges between thebit nodes and check nodes each being associated with a stabilizermeasurement provided by the code; assigning each of the edges in thegraph to a select one of G-number of different edge types; andperforming at least G-number of temporally-separated rounds of qubitoperations, each of the temporally-separated rounds of qubit operationsenacting concurrent multi-qubit operations on endpoints of a subset ofthe edges assigned to a same one of the G different edge types.
 20. Theone or more tangible computer-readable storage media of claim 19,wherein the computer process provides for concurrent measurement ofmultiple stabilizers.